An equivalent criteria for irrationality of $\zeta(5)$

Abstract

Defining a Beukers [1] like integral for $\zeta(5)$ as \begin{equation*} I_n:=\int_{(0,1)^5}\frac{(1-x_3)^n(1-x_4)^n P_n(x_1)P_n(x_2)}{1-(1-x_1x_2x_3x_4)x_5} \ dx_1dx_2dx_3dx_4dx_5 \end{equation*} we prove that for each $n\in\mathbb{N}$ \begin{equation*} I_n= \frac{p_n\zeta(5)+q_n\zeta(4)+r_n\zeta(3)+s_n}{d_n^5} \end{equation*} where $p_n,q_n,r_n,s_n$ are integers and $d_n=\text{lcm}(1,2,...,n)$. We prove that the following are equivalent: 1. $q_n\zeta(4)+r_n\zeta(3)-d_n^5 I_n\notin\mathbb{Z}$ for each natural number $n$. 2. $q_n\zeta(4)+r_n\zeta(3)-d_n^5 I_n\notin\mathbb{Z}$ for infinitely many natural number $n$. 3. $\zeta(5)$ is irrational.